Here we focus on the practical implications [1]}, [2]}, [3]}, [4]}, [5]}, [6]} of the collinear emission of nearly on-shell massive vector bosons, which is the analog in the EW context of the Weizsaecker–Williams emission of photons. The vector bosons \(V\)  participate, as depicted in Figure REF , to a scattering process with a hard scale \(\sqrt{\hat{s}}\)  that is much lower than the muon collision energy \(E_{\rm {cm}}\) . The typical cross-section for \(VV\)  annihilation processes is thus enhanced by \(E_{\rm {cm}}^2/{\hat{s}}\) , relative to the typical cross-section for \(\mu ^+\mu ^-\)  annihilation, whose hard scale is instead \(E_{\rm {cm}}\) . The emission of the \(V\)  bosons from the muons is suppressed by the EW coupling, but the suppression is mitigated or compensated by logarithms of the separation between the EW scale and \(E_{\rm {cm}}\)  (see [2]}, [5]} for a pedagogical overview). The net result is a very large cross-section for VBF processes that occur at \(\sqrt{\hat{s}}\sim m_{\rm {\textsc {w}}}\) , with a tail in \(\sqrt{\hat{s}}\)  up to the TeV scale.
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